5.4 Testing your understanding of the twoserver model
Consider the city depicted in Figure P5.4. In this city emergency repair
service is provided by two response units, prepositioned at (1, 0) and
(1, 0), respectively. Travel distance is right angle, with distance
d between two points (x_{1}, y_{1})
and (x_{2}, y_{2}) equal to
Repairs occur according to a spatial and temporal Poisson process
with
Unit 1's primary response area consists of all points east of the
northsouth boundary line that partitions the city. Unit 2's primary
response area consists of all points west of that line. Note that the
boundary, as drawn, is an equal traveltime boundary line. Whenever an
emergency repair is reported from response area i (i = 1,
2), unit i is assigned, if available;
otherwise, the other unit is assigned, if
available. If neither unit is available, the call is lost (i.e.,
no queueing is allowed).
Repair units travel at 100 miles/hr to and from the scene. Onscene
service time is negative exponential with mean ^{1} = 10 hours. Upon completion of service,
a unit always returns to its home location. Finally, _{o} = 2 x 10^{5} (emergencies per
hour per square mile). Assume the system is operating in steady state.
Average systemwide travel time to an emergency is .
 Using suitable approximations, or exact analysis if you prefer,
find approximate nonzero values for _{1}
and _{2}, where _{m}
workload of unit m.
 The random variable D_{x} is defined to be the
systemwide eastwest (or westeast) travel distance for a random
emergency. Sketch the pdf for D_{x}, again making
suitable approximations.
 Approximately, what is the fraction of dispatches that are
interresponse area dispatches?
For parts (d)(g) only, suppose that _{o} = 10^{2}. This yields new
values for _{1}, _{2}, , and so on.
In considering each of these questions [parts (d)(g)], assume that the
total servicetime distribution remains the same as that assumed in
parts (a)(c).
 Suppose the equaltraveltime boundary line is shifted a distance
( small) toward
unit 1. Which is the appropriate response?
 As a result of the shift, will increase
but _{1}  _{2} will decrease.
 As a result of the shift, will increase
and _{1}  _{2} will increase.
 As a result of the shift, will decrease
and _{1}  _{2} will decrease.
 As a result of the shift, will decrease
but _{1}  _{2} will increase.
 As a result of the shift, we cannot tell which of the four
possibilities above will apply.
 Now suppose that the equaltraveltime boundary line is shifted a
distance (
small) toward unit 2. D_{y} is defined to be the
systemwide northsouth (or southnorth) travel distance for a random
emergency. As a result of the shift,
 E[D_{y}] will stay the same.
 E[D_{y}] will increase.
 E[D_{y}] will decrease.
 The behavior of E[D_{y}] cannot be
determined.
 As in part (e), suppose that the equaltraveltime boundary line
is shifted a distance ( small) toward unit 2. As a result of the shift:
 (_{1} + _{2}) will stay the same.
 (_{1} + _{2}) will increase.
 (_{1} + _{2}) will decrease.
 The behavior of (_{1} + _{2}) cannot be determined.
 Suppose that a third unit is added as a backup unit and is
located at (x = 0, y = 0). This unit is assigned to any
emergency that occurs when both units 1 and 2 are busy and unit 3 is
available. Emergencies that arrive when all three units are busy are
lost. Determine the workload of unit 3, _{3}.
